Nuclear targeting determinants of the far upstream element binding
Binding and Saturation of Nuclear Matter...BINDING AND SATURATION OF NUCLEAR MATTER By PURNA CHANDRA...
Transcript of Binding and Saturation of Nuclear Matter...BINDING AND SATURATION OF NUCLEAR MATTER By PURNA CHANDRA...
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BINDING AND SATURATION OF NUCLEAR MATTER
By
PURNA CHANDRA BHAR3AVA, M.Sc.
A Thesis
Submitted to the Faculty of Graduate Studies
in Partial :FUlfilment of the Requirements
for the Degree
Doctor of Philosophy
McMaster University
September, 1966
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NUCLEAR MATTER
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DOCTOR OF PHILOSOPHY (1966) McMASTER UNIVERSITY, (Physics) Hamilton, Ontario.
TITLE: Binding.and Saturation of Nuclear Matter
AUTHOR: Furna Chandra Bhargava, B.Sc. (Rajputana University)
M.Sc. (Rajasthan University)
SUPERVISOR: Professor D. W. L. Sprung
NUMBER OF PAGES: iv, 141
SCOPE AND CONTENTS:
Using the reference spectrum method the G-matrix elements,
and hence the binding energy of an infinite nucleus is calculated.
Three modern potentials. two of which are hard core and one soft core
are used. This formalism is then extended to a finite nuclei and
spin orbit splittings around some closed shell nuclei are calculated.
Both for binding energy and spin orbit splittings fairly good agreement
with experiment is obtained.
(ii)
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ACKNOWLEDGEMENTS
It is a pleasure and a privil~.ge to thank
Dr. D. W. L. Sprung for his guidance, encouragement and constructive
criticism of this, and other work throughout my stay at McMaster
University.
I am thankful to Professor M. A. Preston and Dr. R. K. Bhaduri
for several useful discussions. Thanks are also due to Dr. R. K. Bhaduri
for having supplied to me the oscillator wave functions and the shell
model density distributions; to Dr. c. N. Bressel and Mr. R. V. Reid
for supplying their potentials before publication; to Mr. P. K. Banerjee
for having prepared part of Table X and to Mr. W. van Dijk for having
done some phase shift calculations for me.
For computing work I gratefully acknowledge the assistance of
Mr. D. J. Hughes, Mr. R. Deegan and Mrs. Ann Taylor during the initial
stages of this work. The work was done on the I.B.M. 7040 computer,
and the ready co-operation of Dr. G. L. Keech and his staff is appreciated.
For financial assistance I am grateful to the Physics Department,
McMaster University, and the D. G. Burns Scholarship.
La.st, but not least, thanks are due to Frau Maureen von Lieres
for typing this thesis so well.
(iii)
http:privil~.ge
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Chapter I
Chapter II
CHAPTER III
CHAPTER IV
CHAPTER V
APPENDIX A
APPENDIX B
APPENDIX C
TABLE OF CONTENTS
Introduction
Approximate Methods for Solving the
G-Matrix
Description of the Calculation
Discussion
Spin-Orbit Force
Coupled States
Potentials
(iv)
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CHAPTER I
INTRODUCTION
Nuclear Matter is a hypothetical nucleus of infinite size.
It contains an equal number of protons and neutrons, with no coulomb
interaction present, and because of its infinite size no surface
effect. Obviously this does not occur in nature. Still in a sense
nuclei do resemble such an abstraction when seen from the liquid-
drop model point of view. In this model the nucleus is supposed to
have a central region of uniform density of particles, with a surface
where the density falls away rapidly to zero.
There are two lines of evidence to show that this is a
fairly accurate picture of a nucleus:
(1) Assuming this model the Bethe-Weizsacker semi-empirical
mass formula can be obtained:
The first term comes from the uniform density of the nucleus. Unless
it is near the surface the particle sees the same surroundings every
where, and has the same energy.
1
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The second, third and fourth terms are respectively to
take into account the surface effect, the coulomb effect, and the
symmetry energy. There is also a fifth term to allow for the
pairing energy. The parameters in this equation have been adjusted
to give a good fit (± 500 K~V) to the general tendencies of nuclear
binding energies.
(2) Electron scattering experiments by Hofstadter show that
nuclei A ) 16 have a uniform central density of about .168 nucleons
per r3 which falls from 0.9 to 0.1 of its maximum value over a distance
of 2.5 F.
For an infinite nucleus the binding energy per particle is
given by the first term -a1 in Equation (I-1). Empirically this
value has been determined by Green, and Cameron (1) to be between
-15.8 MeV and -17.2 MeV. The goal of nuclear matter theory is to
obtain this value from theory at the correct saturation density.
Thus, the model under consideration is an infinite number of
nucleons filling co-ordinate space. Due to translational invariance,
the single particle states are piane waves. Since nucleons are Fermi
particles they will occupy the lowest available state according to the
Pauli's exclusion principle, four nucleons going in each state. The
filled states will form a sphere in momentum space of radius Pr (Fermi
momentUJ)l) extending to the highest occupied state corresponding to the
most energetic particle (Fermi energy). Generally we use 11 = 1 and
talk in terms of the wave number kF =p~ as the Fermi momentum. To deal with this interacting Fermi gas we must use a many body formalism.
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The details of the many body formalisms have been given
in several places (2) and need not be repeated here in any detail.
It will be sufficient to mention that the most usual procedure is to use
the perturbation theory and seek an effective one-body potential from
a given two particle interaction, as in the Hartree•Fock approach.
Normally this method would be excellent, but nuclear forces are known
to be strongly repulsive at short distances a' ·'. the Hartree-Fock method
would fail for them, because the matrix elements of the potential
would be very large, if not infinite. The solution to this difficulty
is in Brueckner's theory of nuclear structure. This theory has also
been dealt in detail in several places (3). Essentially this
procedure is an extension of the Hartree-Fock method to strong short
range forces by allowing the interacting particles to interact any
number of times before they return to the Fermi sea, so that the
resulting effective two-body interaction is given by a matrix 'G'
which is the "sum" of a series of matrix elements of 'v', which
alternate in sign. The matrix elements of 'G' between the states
of two particles of initial momentum m1, n and of final momentum1
m, n are then written as
1- ~ [ ~f (I-2)
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or more formally
(I-3)
where Q is the Pauli operator which requires that the intermediate
states be above ~· e is the energy denominator defined positive
and given by
(I-4)
It will be seen from the definition of the G-matrix in
Equation (I-3) that its structure very much resembles that of the
scattering matrix t or K of two free particles. Actually it is the
same, except for the operator Q which prohibits transitions to the
occupied states which do not exist in the case of free scattering.
This then is the main difference between free scattering of two
nucleons and that of those in nuclear matter. The presence of this
operator makes the solution for G-matrix elements very complicated and
various simplifying means have to be developed. The other difference
is that the energy demoninator includes binding effects of the nuclear
medium.
One of the most important applications of the 'G' matrix
formalism has been to derive the effective single particle potential
due to other nucleons e:Kperienced by a single nucleon in the Fermi
sea, and hence its binding energy to the system. A simple extension
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of Hartree-Fock procedure to the 'G'-matrix shows that a particle
of momentum km experiences a potential U(m) given by
ll ::: L ( mYJ j G / rnn-nm) n· E ·) -- ( K.£.') + ~ ( P. E.) 112.. ~~~) (I-6) = + l.. UllYI)
:;2.a IYl
2.~ 2..3 i) F I= u l Yl'l)+5 ~a fY1
A study of the binding energy characteristics according to density also
leads us to a saturation density which can be compared with the nuclear
density inside a large nucleus like Pb2o8. In the next three chapters
we give the details of our approach to this aim. In Chapter II we
give the simplified methods to calculate the G-matrix elements and discuss
the physical ideas of the origin of different terms and the contributions
of different regions of the two body potentials. Chapter Ill is
devoted to techniques 0£ calculation. In Chapter IV we present the
results of all this exercise and come to the conclusion that the hard
core potentials give better results than before an4 that the soft core
potentials may indeed lead to the right b5nding energy and saturation
density when the three body clusters are taken into account. Only
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the lowest order three body clusters are removed in Hartree-Fock
type approach. Very briefly considering .them merely represents
the fact that for very short range strong forces like those in a
nucleus, more than two particles may interact before returning to
their normal situation in the Fermi sea. The proc·edure for
taking them into account was given by Bethe (4) and is briefly given
in Chapter II.
So far we have only conF~ dered the application of the G
matrix to an infinite nucleax system where only effective central
forces exist. A finite system has a surface and therefore the
tensor, spin-orbit and other forces can come into play. The extension
to such a system was done by Brueckner, Gammel and Weitzner (5) under
the local density approximation. In Chapter V we have extended the
G-matrix calculations to study the spin-orbit splittings in a few
closed shell ±1 nuclei and obtain values in reasonable agreement with
experiment. We also find that the spin-orbit force peaks about 0.15 F
inside the half density radius and go on to argue that the spin-orbit
force is about .15 F inside the central force, confirming the recent
works of several people (36). We also show that the two body spin
orbit force obtained directly from the two body potential by itself is
sufficient to account for the splitting of the levels.
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CHAPTER II
APPROXIMATE METHODS FOR SOLVING THE "G" MATRIX
There are essentially three ways to solve for the
G-matrix. The first one was given by Brueckner ( 8) himself.
It is a direct approach to the problem and at least in principle
should lead to a very great accuracy. But the procedure is very
lengthy and complicated and still requires an approximation for the
core region. The other two procedures make approximations to the
G-matrix, but give more physical insight into the problem and simplify
the calculation. These are the Moszkowski-Scott separation method
( 9) and the Reference Spectrum method (10). The Moszkowski-Scott
separation method has been modified by Bethe, Brandow and Petschek (10)
(henceforth referred to as B.B.P.) to improve its accuracy and is then
called the modified Moszkowski-Scott separation method. The methods
were developed to give greater physical insight into the theory.
In particular, one wishes to understand the role of the exclusion
principle, the hard core, and the deviation of the energy denominator
e(k) from a simple effective mass form. Earlier methods treated
these things in various ways, but nothing was very clear.
In the Moszkowski-Scott method the long range part vi of
v and the short range part v were separated out, such that the s
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attraction due to v was completely balanced by the hard core s
repulsion and these together produced no phase shift. The distance
'J' at which this separation occurred was called the separation
-0.istance. Moszkowski-Scott then calculated Gs and G,.e as if vs and
vt were independent of each other and showed that the correction
G - (Gt +Gs) are 10 to 20% of the main term. v~ is relatively weak
and of long range and hence could mainly induce transitions within
the Fermi sea which are prohibited by the exclusion principle. Thus
Gt~ vt - vt ~ V,e• The second order term was small. This also
meant that the wavefunction ~ differs little from the uncorrelated
wavefunction ¢ in the region vt. Thus " tf.," corresponded to the
"healing" distance introduced by Gomez 9 Walecka and Weisskopf (11).
For v the exclusion principle is of little importance, since it s
mainly permits transitions only beyond the Fermi sea. Thus the
Moszkowski-Scott separation method brought out the role of the
exclusion principle and other things more clearly and understandably
from the physical point of view. To improve accuracy this method
was slightly modified by B.B.P. We shall talk of it later.
The Reference Spectrum Method
The understanding of Nuclear Matter given by Moszkowski and
Scott was made use of by Bethe, Brandow and Petsche~ (10) to develop
the "Reference Spectrum Method". Here one can calculate the 'G'
matrix by a procedure which is more accurate and simpler than the
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separation method. They showed, first, that given two types of 'G'
matrices GN and GR which differ from each other only in the intermed
iate state projection and in the energy denominator, they are related
by
(II-1)
where PR and PN contain the intermediate state projection operator
R Nand the energy denominator for each of the G-matrices G and G •
Now if GN is the nuclear G matrix (Equation I-3) then PN = Q/eN
where Q is the Pauli operator prohibiting transitions to the filled
Fermi sea. Then
(II-2)
B.B.P. tried to find a GR which is such a good approximation to GN so
that the second term in (II-2) is small. This would then permit the
replacement of GN in the second term by GR. The Moszkowski-Scott
Rmethod, and studies by Kohler (12) required that G should take a good
approximate account of e N
(k) for k ) kF and of Q prohibiting transitions
to states k { kF. But simplicity and ease of calculations required
that Q be ignored and eN(k) be replaced by a quadratic form Ca1 + a2 k ),
because this is equal to the operator (a - a V 2) • B.B.P. re1 2
investigated Brueckner's single particle energies and found that for
~ )) kF' UN(h), the single particle potential energies are very nearly
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quadratic in kb. Thus to a good approximation, one could replace
for kb ~ 1.5 kv UN(b) by a "Reference Spectrum" potential:
(II-3)
The intermediate state-particle energies in the "Reference Spectrum"
approximation then becomes
:::. A -t (II-4)
where
2.
=- Tl -1
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D ~21F2 Thus E 0
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where eR is given by (II-9). Introducing the two-body correlated
wavefunction 'f'and the uncorrelated wavefunction ¢ satsifying
(II-12)
i.e. defining
(II-13)
we can write
- ,,,/... I 7.)- W~'-/-' - el?. .., (II-14)
which with eR as given above (Equation II-9) gives the reference
wave equation
(II-15)
where
(II-16)
is the wavefunction distortion. Equation (II-15) is the fundamental equation
of the reference spectrum method. When expanded in partial waves
for uncoupled states it leads to equations of the form
(II-17)
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We use the convention of B.B.P.:
(a)
(b) (II-18)
(c)
and
(II-19)
y 2 for the occupied initial states is given above (Equation II-10);
for particle states B.B.P. have after considering the off-energy shell
effect that
2. 2. 2 y (II-20)( 31::> - 0 · {,) ~ F + 3 -ko
Comparing (II-12) and (II-17), and making allowance for an infinite
hard-core of radius c, we find
(II-21)
471 [ 2 l.. Jc. '1..rn*-k~ C-Ao +1 ) 9i- c-kokJ d.k
0 (II-22)
001
+ $" (9i- - 9:1: )Jc + m* 1C9t. - W") V-ULrJz> &IA]
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with
(II-23)
Here
(II-24)
where (-J ·l+I . l+J •rh =~ (,
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00
t 3 L (d.L.+ 1) 1 ~1.Jl
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After the reference GR matrix is known, the correct GN
matrix can be determined by solving the equation (see Equation II-1)
(II-26)
which was derived in their Appendix A by B.B.P. Assuming that GR
is a good first approximation to GN, this may be solved by iteration
The terms on the r.h.s. we refer to as the second, third, ••• order
corrections to GR. The second order terms were estimated by B.B.P.
and by Brown Schappert and Wong (13), but the third order term was not
seriously considered previously. The calculation of the second
order term, LJ a proceeds as follows. Consider an interacting
pair of particles, denoting their momenta kb, k by P ± k , in a A;. m o
definite spin, isospin state. The interaction conserves P, S, and T,
so the intermediate state sum is only over relative momentum k'.
The energy denominators involve the kinetic and potential energies of
the initial states (,e and m) and the intermediate states (a and b)
of the two particles:
fl.' N T(./
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In units of 112/m = 41.469 MeV - ~we have
(II.29)T ( -k.e } + T ( -km)
u (-/
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where
Mand the average of 1-sT over spin and isospin states is
(II.36)
is a spherical Hankel transform of the wave function distortion X ~'L
in the coupled state with total angular momentum J, taking the solution
with dominant orbital momentum L and looking at the L' component. For
example, tne deuteron (J:l) is mostly S-wave, so L::O and the large and
small components are labelled by L' =o, 2 respectively. For uncoupled
states, only the index L is required. The average =fo..,. is of course
all one requires in binding energy calculations.
In Equation (II-33) the angular integration has been performed.
This involves an angle-averaging over the Pauli operator Q and thus in
Equation (II-34) it is this angle averaged operator which occurs.
B.B.P. considered the case P =0 so Q was simply a step function at We find that by taking Q (P0 o-, k') the second orderk' =~·
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corrections are reduced by about 10%. The angle averaged
Pauli operator is given as
0 I
=
=
In our calculations the greatest part of the second order
correction comes in the coupled 3s1 - 3n1 partial waves, where the tensor force is important. In fact the sum of all other second
order corrections is virtually zero. The small component X ' 20 of the "deuteron state" does not occur in the first order GR calculation,
but does enter Equation (II-35) for the corrections.
fourier transform of the major, S-wave component is negative at small k
(from the long range attraction) and positive at moderate k
several kF, due to the short range repulsion. Empirically it is
small near ~ due to these compensating effects. F201 , however is
the transform of a function of one sign, zero inside the core.
~20 is negative at all k upto about 6kF, and takes its maximum value near 2kF, as in Figure 1. Since it is large near the Fermi surface
it makes a big contribution to both the Pauli and spectral correction
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terms. Because so much of the correction came from this one state
we evaluated the third order correction term, and found it to be
reasonably small, but not negligible.
The evaluation is similar to that of the second order term.
Taking the matrix element of b,. (3)G between states of definite P, k ,0
S, M and T we have
(II-38)
( I - .fl.~) /. =L_ ( ~·M'/ E. (-f/J/*'M'>s~)
The wave operator s!' has the property
(II-4o)
Rt (k) was defined in Equation ( II-34) and t = E. /e (k). There are seen to be two intermediate state sums, but as before P, S, and T are
conserved by the interaction. Equation (II-39) is written for triplet
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spin states, but the singlet case is an obvious simplification.
Using Equations (6.28), (6.27) and (6.31) of B.B.P. we write
(for S =1, T =O):
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In view of these relations, Equation (II-38) gives
00 131 ~ ( ¢s; / i'J GJ cPs;) =1d f/ d f/' i (-/:,!'1 ct~f/) JJ;(YJ. (~'-7io):J):.m"(-*"~'J/J F. :J'
L,._L, ( /< ~ f
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is sufficiently complicated that we considered the Pauli operator
to be a step function ink', k11 ; i.e. we took P =0 in Equation (II-37).
Effects of Three Body Clusters
For occupied states, the single particle potential energy
is defined as
{) (m}""' L
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B.B.P 0 showed that this insertion cannot be put on the energy shell
by generalised time ordering. By a simple estimate of this effect,
B.B.P. derived the estimates Equations (II-10) and (II-20) for the
reference energy denominators. It is clear that Y 2 for particle
states (b-n interaction) is much larger than for hole states (m-n
interaction) and this has considerable effect on raising the potential
energies of particle states U(b). In short, Equation (II-49) is also
used for U(b), but G(b,n) is calculated on a different basis.
The motive in defining a potential energy of particle states
is to reduce higher order terms in the perturbation series. In the
above, only the third order "bubble" diagrams were considered, because
the third order "ring" diagram (Figure 2D) was known to be small. ,
More recently it has been realized that all diagrams containing only
three occupied state lines should be considered together as constituting
a three body cluster. Rajaraman ( 16) first pointed out that all
the diagrams of third order in G could be grouped together, considered
as insertions in a particle line b, and therefore cancelled by a
suitable definition of U(b). Essentially there are two distinct third
order diagrams, the "bubble" and the "ring", plus all their exchanges.
Rajaraman arrived at a very simple prescription, namely that Equation (II-49)
is used, but summing only. over ~ partial waves of G and using
statistical weight 1 in place of 3/4 as usually applied. This was done
in Razavy' s ( 17) calculation.
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Rajaraman ( 18) also pointed out that there are diagrams
of all orders in G containing only three occupied state lines. These
are, therefore, of the same order in the density. Bethe (4 ) derived
a Faddeev equation for the sum of all these diagrams and constructed an
approximate solution. His result is that the sum of all three body
cluster diagrams can be written as the lowest (third) order diagrams
alone, with the middle G-interaction modified to (G.f.). The
"suppression factor" f(r) is one at large r, but falls to about 1/3
inside the hard core (Figure 3 ). It was, therefore, suggested by
Bethe that the three body clusters be incorporated in the calculation
by defining
U (4)::: L (411/[Gf]/tn-n-t) (II-50) -B.,,
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still applies. This seems clear because the complete sum is a
weighted third order diagram alone.
Bethe estimated f(r) to take average values .384 inside
the core and .86 outside. The effect of f(r) is thus to make the
energies U(b) much more attractive than formerly, since the repulsive
contributions are relatively more suppressed. It was estimated (19)
that the three body clusters contribute of the order 1 MeV to the
binding energy compared to about 10 MeV repulsion when the third
order alone was considered; this is the source of the improvement
in agreement with experiment.
In our calculation we took somewhat better account of the
suppression factor f(r), using Equation (II-51) and (II-52), than in
Bethe's paper. For the hard core potentials we used his numerical
values of f(r), scaled to the appropriate core radius. In principle,
f(r) should be evaluated for the actual potential used, and thus our
work is subject to a correction on this point. M. w. Kirson (20)
is considering this, while B. Day (21) has recently improved on Bethe's
solution of the Faddeev equations. For the soft core Bressel potential,
we evaluated f(r) on the same basis as Bethe's hard core function.
The relevant equations are (4.4) to (4.6) and (3.24) of reference (4 ),
which we reproduce here:
·(II-53)
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(II-54)
(II-55)
UI~ - I - '°S,::z. (II-56)
ri,2- Ll,3+11,3 u,:z-"'1:J 3 ( u,2- r-u,3 -J l/12 u,3 )
ul:L u,3 .,_ llu. ll23 + ll3, ll3:i_- r1 u,2 ll23 ll3, (II-57)
The subscripts denote the argument of each function, so u12 =u(r12)
etc. Y/ and 'f are the wave function distortions (more precisely
l} · ¢ and 'S· ¢ are) for a typical hole-hole and particle-hole inter
action respectively. One requires these two situations because the
initial and final stage in any three body cluster is the interaction
of two particles in occupied states. The remaining stages involve
excited particles. Equation (II-57) is Bethe's approximate solution
of the Faddeev equations. ~(l) can be interpreted as the part of the
three body wave function after any number of two body interactions,
when particle l does not take part in the final interaction. The
energy due to three body clusters is
W = j G ( ~z.3} F ( k.u) d 'T,,.3 (II-59)
while by considering the third order only one would have
(II-60)
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Using (II-55) it is obvious that
(II-61)
which leads to the interpretation stated earlier.
In Bethe's calculation for f(r) several approximations were
made. Only the short range part of the interaction was used, the long
range part removed by a modified Moszkowski-Scott ( 10) separation.
This is helpful because~ and "$are then positive functions, and
justified because only the strong short range force can make appreciable
contribution in high order diagrams. This is done here. Next ')'(r)
was taken to be a real function of r only. Actually,
(II-62)
where X = ¢ - 11' is the wave function distortion in Equation ( II-16).
Clearly 'f is in general a complex function of r and e. If ¢ and 7C
are expanded in partial waves, and 'f in a series of multipoles,
f(.I;) :;:: L (:ll-t-t) .iL ~(Ji:)~ ( UJ:j(}) £_
the 'fL may be determined by term by term comparison. ~ L:O is the
angle average of ~ and it is only in this approximation ( -S = ~o ) that
~ will be a real function only of r. Still considering only spinless
particles,
(II-64)
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This agrees with (II-62) in giving 'f = 1 inside a hard core.
Finally, Bethe did not calculate 'S (r) for any particular potential,
but adopted a simple and reasonable quadratic form for it.
In our calculation we constructed .X(r) for the 1s Bressel 0
potential using the Modified Moszkowski-Scott method. This was done
for the two representative hole and particle cases. -rJ and "$
could then be taken from the L =0 parts of either (II-62) or (II-64).
Since Equation (II-62) is too large near a zero of ffi
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Modified Moszkowski-Scott Method
As mentioned in the beginning of this section in
Moszkowski and Scott separation method beyond the separation distance
dF the wavefunction of two free nucleons interacting via vs goes
over to the unperturbed wavefunction. B.B.P. modified this condition
to the effect that the wavefunction in the reference spectrum goes to
the unperturbed wave function beyond d R" This is more accurate than
the MS condition because the reference G-rnatrix is much closer to the
actual G than the free nucleon G-matrix. This condition means
R ( -ko,Jc.) ;::: oX. 05 (II-65)
or equivalently
(II-66)
Since /( and }..' are continuous, condition (II-65) is equal to
Ix ( d- ~) =- x cot-o =-o (II-67)
B.B.P. then go on to show that up to the terms in second order
(II-68)
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where K is a small quantity given by
(II-69)
where 00
~ 1( -F- '/ / - JI_~ / ~D) = ~ ~ 19-L { f/h) X/. 5 (-ko, 'l) d k ::f 0 -k ~0 o (II-70)
I
f.? :;; 0 :::
Proceeding just as we did in the Reference Spectrum Method, we can
easily obtain
(II-71)
(II-72)
for L = 0 wave alone.
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The third term in (II-68) is given by
00
N -J 1
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CHAPTER III
DESCRIPTION OF THE CALCULATION
(a) Reference Spectrum
The first order or reference spectrum calculation involves
the solution of the reference wave equation (II-17) and the evaluation
of the integrals in Equation (II-21). To simplify the discussion we
consider singlet states and relegate the triplet equations to Appendix A.
Equation (II-17) may be written as
(III-1)
(III-2)
For a hard core potential, uL vanishes for r ( c so /( L is equal to J--L.
We solve Equation (III-1) for r ) c, subject to X -7 0 (heals) at
large r. In practice we assume that v(r) is negligible for r greater
than some large distance d. Beyond this distance,
(III-3)
33
-
with Na constant and 'FIL a decaying Hankel function as in Equation (II-24). Explicitly
'flo (~) ""'-x
e
'II, {/r.) :: (1-1-~) -x e (III-4)
~+-1(}~) --.1. l +-I
x '#/.. {1r_,J -f '# UcJ L-1 x :::. Yk
Giving a name
9>j_(k}~-- "- ~~ d.k
- y (III-5)ro
to the log-derivative of 9/L' we can write the two point boundary
conditions on Equation (III-1) as
(III-6) A:: d
In our work we generally used d ~ 7.lF. This problem is conveniently
solved by the method of E. c. Ridley (22 ) as suggested by Razavy ( 17 ).
Actually, Razavy used a variant of this method starting the integration
at r = c. By starting instead at r = d the procedure is greatly
simplified, as described here.
In the Ridley method, auxiliary functions s and w are
introduced, so that Equation (III-1) is factored into three first
-
35
order equations,
(III-7)
(III-8)
(III-9)
with (III-10)
(III-11)
The boundary condition at r = d is satisfied (see (III-9)) if s(d) = r L and Ct.J (d) =O. Using these initial values, (III-7) and (III-8) are integrated inwards to r = c, constructing a table of s and (1..1 by the
Gill-Runge-Kutta method ( 23). Finally, using /((c) from (III-6) as
initial value, Equation (III-9) may be integrated back out to r = d,
completing the solution. The mesh here is double that used on the
inward integration, so s(r) and UJ (r) are available at the mid-points
of the steps.
The values of /((r) are now used to construct the integrals
in Equation (II-24) for r ) c. These may be considered spherical
-
Hankel transforms, and conveniently evaluated by an extension of
Filan' s method ( 24 ) • This is a generalization of Simpson's rule in
which the wiggles of sin (qr) are exactly accounted for, so the mesh
required is determined by the smoothness of X (r). Thus, for all q
of interest we can use the same table of X(r). Our computer routine
FILON assumed X(r) to be tabulated in blocks of 24 steps, with the
mesh size doubling between blocks. For the hard core potentials we
used H = .04 F outside the core, and three blocks in all. For the
soft core Bressel potential we used a fictitious hard core of O.OlF
as a device to start the integration. We fitted two blocks exactly
inside the square core region and three blocks outside, being careful
to pick up the very deep value of v(r) immediately beyond the square
potential edge. For the region r ) d, so far neglected, we added a
correction. Assuming as before that v(r) is very small, it is easy
to derive the approximation
JJoO
JJ. ( /!0 Jc) XJ. ( -P-o-t:) ~ ~ (III-12)
For r ( c '"XL = i L and hence the core contribution is given by
(III-13)
-
37
The Reid hard core potential, given separately for each
partial wave, is available only for the following states:
For sake of comparison we calculated the binding energies for these
very states for the Hamada-Johnston and the Bressel-Kerman potentials.
For higher partial waves we evaluated the OPE.P contribution
in Born approximation as first used by Razavy. For this purpose we
assume that only the OPEP is effective for these waves and replace
u1 by JL as in Born approximation in (II-25a). This procedure is quite good since the centrifugal barrier allows these partial waves
to be effective only at large distances for the energies in nuclear
matter and the wave function is not too much distorted. The detailed
procedure is this: We first assume that the OPEP acts in all partial
waves and then subtract out the ones which we have taken into account
more accurately. The OPEP is given by
0·0758
2 µ c :::: PioN M11ss
Since the tensor part averages out to zero when we sum over all the
J components of any partial wave we have writing
-P .!c -e (III-15)
-
s
-
39
I
16
(III-20)
Similarly if only the even waves from amongst those given on page 37
are considered the OPEP contribution is
(III-21)
The procedure of calculation was as follows: For each ~
reference spectrum parameters /1 and m* were chosen. After some time
it became clear that one can choose /'). = o.6 and m* (always near 1.0)
was determined by making the reference spectrum cross the "nuclearn
or computed spectrum at some ~ ~ 3 kF. We chose seven values of
k0/kF =0.1, 0.2, o.4, 0.54 0.775, 0.9 and 1.0 and for each k
0
R 2constructed G (k ). The y for given k is determined by Equatioh (II-10).0 0
In the first order calculation, GR depends only on k , not P, so 0
(III-22)
-
4o
and the average potential energy of occupied states is
U ~ ~ L U(rn) m
-
41
where k is the average relative momentum between occupied states. 0
Since b (2 )G is not exactly a quadratic function of k , this is not 0
exact. Later on we evaluated /;:).( 2 )G at seven values of k and 0
integrated accurately to find the change in binding, as in Equation
(II-49). As shown in Table I, the one-point approximation is very
good, with an error of order 5%. The angle averaged Pauli operator
Equation (II-37) with the P appropriate to k was used in all cases. av o
Compared to setting P = O, (so Q is a step function), this decreases
the corrections by about 10%.
We made one slight error in our use of Equation (II-32),
2by using ~(k') in place of the argument IP 2 + k' • For large~ av
k' this makes extremely little difference and we feel there are
greater uncertainties than this in the nuclear spectrum.
With P ~ 0 the integration in Equation(II-33) breaks up
into four regions, two Pauli and two spectral. The three short
regions were covered by 9 point Simpson's rule, the longer region by
21 points. In our most accurate evaluation of the spectral correction
we found some difficulty with taking k =kF. Due to the near 0
Ncontinuity of the nuclear spectrum at ~' e tends to zero at the
Fermi surface. However P~~ also tends to zero in this limit, so Q
is a step function. To evaluate the spectral correction we took the
integrand at k =~ as a polynomial extrapolation from the values above ~· This is probably justified, and was the only numerical problem
faced.
-
42
(c) Third Order Correction
The third order correction was evaluated from Equation (II-46)
and (II-48). In this case we took Q to be a step function, to keep
the work to a mininnun. Further, we used the "one-point" approximation
(III-25) to estimate the change in binding energy. It would be a very
lengthy task to evaluate ~(3)G at several initial k • Considering0
the coupled 3s1 - ~l waves, J = l and it turns out there are eight
combinations of L-values in the sum (II-48). These are shown in
Table :LI , in order of importance. The region of integration is
shown in Figure 5 • The upper limit was set at 6~ which we believe
is adequate. The Pauli operator Q divides the square into four
regions, so with the eight terms there are thirty-two contributions
to the corrections. In practice only one or at most three of these
contributions was significant, the others being several tj_mes smaller.
This gives us confidence that the result obtained is meaningful, and
indicates also that the series of correction terms is converging
absolutely.
Simpson's rule was used in the integration, with eight
points below kF and 21 above. According to Equation (II-48) we must
have 29 Hankel transforms of X (k , fl. ) for the table of ii L(k", k ) • 0 l 0
However, the final term in the sum ~ould require 29 transforms of
each 29 reference wave defect functions X (k', r). This would be a
large amount of computing and we made the following approximation.
-
--
We chose seven representative values of k', k'fk:r = .22, .56, .775,
.937, 1.5, 3.125, and 5.125. We found XR for each of these and
for each made a table of the transform. During the integration
the required F(k", k') was taken from the correct row of the closest
available k' (above or below kF as required). Some comparisons
given in Table III show that even with only two k' - columns the
results agree quite closely with each other. We believe that our
evaluation of L)(3) U is good to better than 10%.
We also calculated the third order correction for the
1s state for a few representative cases. It was of the order 0.2 MeV,0
so completely negligible.
(d) MMS Method
Since the corrections to the Reference Spectrum are most
important in the S state we proceeded to apply the HMS method only in
1the s state, the 3s1 being taken care of by the third order correction. 0 Equation (II-75) then becomes
or (III-26)
The boundary conditions are given as
Xo .1, ::: lo Ih~ c (a)
::: 0 Ik,, cl--~ f"--') 0 (b) (III-27)
"
-
44
The equation is again solved by the method of E. c. Ridley, but
now, because the separation distance is not known at first, we must
use its variant, as was done by Razavy.
Equation (III-24) is again factored into three first order
equations:
ds - - (1+ 5fJS) (III-28)-cJ.>t
cia; - - 5(9W--fi)- (III-29)d-'L
dll :: r; ( 5u-1-w) --Ii (III-30)dh
:: SU+WXo-6 (III-31)
- u (III-32)and
::: - ( 1' 2- yr? * l!.i )-f where (III-33)
(III-34)
The equations are similar to (III-7, 8, 9, 10, 11) if we identify
5 = I .J (III-35)
and
-
45
The boundary condition (III-27a) is satisfied if we put S = 0
and hf = Q (kc) at r = c (Equation III-31). With these initialq 0 0 values we integrate outwards for S and W, constructing a table for
them in this operation, till bJ = O. Here we then put U =o, thus
satisfying the other two boundary conditions (III-2lb and c), and
integrate inwards for U, constructing a table for it too.
is then calculated from Equation (III-31). The evaluation of
(k /G~/k ) is now given by Equation (II-71).0 s 0
The evaluation of the other terms in (II-68) is quite
obvious from (II-72) and (II-74). The effect on potential energy
of the correction terms was evaluated as in the Reference Spectrum
Method.
-
CHAPTER IV
DISCUSSION
We have carried out the nuclear matter calculations for
three modern potentials, two of which have hard cores (Hamada-
Johnston ( 26 ) and Reid ( 27)) and one has finite or "soft"
repulsive cores (Bressel-Kerman ( 28)). The potentials are des
cribed in Appendix B. The results for binding and saturation
are su..rnmarised in Tables IV and V • With the inclusion of three-
body cluster effects, both binding and saturation are closer to
the experimental values than in previous calculations. The result
for the Bressel Kerman potential shows that it is feasible to hope
for agreement with the experimental values of about 16 MeV per
particle at kF =1.4 F-1 • The main surprise is the 2 MeV difference between the Reid
and Hamada-Johnston potentials, and the four MeV gain of the soft
core Bressel-Kerman potential over its parent, Hamada-Johnston,
{Figure 6 , Table V ) • By referring to Table IV, it can be seen
that the gain in binding of Reid over Hamada-Johnsto~ is in the 3n2 and the net 3p states. Evaluation of the phase shifts for these
3potentials shows that in the n state, Reid's potential is more2
46
-
47
attractive, its phase shift being about 10 to 15% greater (and
agreeing better with the most recent phase shift analyses). Since
the average potential energy due to 3n interactions is about 15 MeV,2 we can expect Reid to give about 1 MeV extra binding on this account.
A similar contribution from the 3p states accounts for the difference.
The Bressel potential gain in binding is a more subtle effect. First
of all, softening the core does give a gain of about 1 MeV binding in
each of the S-states at a given density. The comparison of Bressel
to Reid rather than Hamada-Johnston is undoubtedly the fairer one,
since both Bressel and Reid fit the most recent data and we have just
seen that differences of a few MeV can easily arise from fitting
different data :0lections (i.e. giving different phase shifts).
The remaining small gain in binding is due to saturation occuring at
a higher density. This will be brought out more fully below.
Our work shows that at least for the hard core potentials
the third order correction to the reference spectrum approximation is
significant, being of order 1 MeV. This can be seen in· Tables VI
and VII. In the soft core Bressel potential it is much smaller and
generally unimportant (Table VII:O. This seems confirmed by pre
liminary calculations with another soft core potential. It can be
seen in Tables VI and VII that calculations made from different
reference spectrum parameters l1 and m•, if carried to third order,
-
48
agree remarkably well: within .5 MeV in all cases. This is in
spite of up to 5 MeV differences in the first order result. We
have also calculated the 1s state by the modified Moszkowskio
Scott Method, obtaining similar values as shown in Table IX •
This agreement gives confidence that the reference spectrum series
is converging to a meaningful GN matrix. It also indicates that
only a modest degree of self consistency is required between ~ N
and U • Of course, if /1 and m• vary too greatly from reasonable
values, the individual correction terms become large and their
relative inaccuracy more dangerous. For example, in Table VII
we calculated one case for the situation U(b) =0 m• =1, with
self consistency in fl ) which gives fJ.. =o.86. The second order corrections are huge (spectral = -23 MeV) and could easily be
inaccurate by one or two MeV. The final result is -13.2 MeV
binding at kF = 1 0 36 F-1 , differing by about 2 MeV from our best
value. A substantial third order 1s state contribution could 0
contribute to the discrepancy.
It could be argued that once one decides to define U(b) =0
the spectral correction should not be included at all. Rather the
contribution of three body cluster diagrams should be included as a
perturbation. Removing the spectral correction would reduce the
binding disastrously, and it is unlikely the three body cluster diagrams
can account for the discrepancy. It appears that at least two or
-
three MeV binding are arising in our calculation from the iteration
of three body clusters in higher order diagrams. This comes in when
we define U(b) to be something' non-zero. This point requires further
study.
To complete the point we began, evaluation of the third order
reference spectrum correction is quite simple, and is apparently an
adequate method to get an accurate result from the tensor force.
Generally speaking, we see in Tables VI, VII and VIII a large
compensation between the second order Pauli and spectral corrections,
with a small net result. This is true until Li and or m* is too
far from rough self consistency. We have already noted that there
is no such cancellation between parts of fj,(3)G. Thus from the point
of view of absolute convergency it is fair to say the potential energy
contributions from first second and third order are in the ratio
80 : 20 : 2, which is satisfactory.
It should be noted that decreasing [). makes the Pauli
correction large, while increasing D. makes the spectral correction
large. The Pauli correction is determined mainly by the magnitude
2 RA small Lb: 0.5 makes Y small, so X (r)
has a long tail. This makes F(k') large at small relative momentum.
The spectral correction is sensitive to eR - eN in the region above
but near ~· This difference is nearly proportional to tJ. •
Empirically, the net second order correction is small for L1~0.6,
which makes this the most useful value to employ for reference spectrum
calculations.
-
50
Many details of the average potential energy per state are
apparent in Table IV , and are more easily seen in Figures 7, 8
and 9 • Over the range of densities displayed, the average
kinetic energy T is roughly linear in ~; of course it is really
1quadratic. The potential energy due to S interactions is seen 0
to be parallel to the kinetic energy (-T) line. This shows the
1effect of the short range repulsion, in cutting down the S 0
2contribution from a kF3 dependence to only kF • The states with
J } 2, interestingly, all have about the same dependence on kF and
vary more rapidly. Thus, the many accidental cancellations which
occur, hold over a wide range of densities. The 3p states combine
to have a very small net result, consistent with the usual inter
pretations that the P state force is primarily spin-orbit. The
higher partial waves, here given by OPEP, make a substantial repulsive
contribution which is highly density dependent. The 1P1 state
repulsion is slightly stronger. Together these odd states roughly
cancel the large attractive 3n2 contribution. The remaining attractive state, 1D2 is not shown as it lies just between the lines (-OPEP) and
(-l P1 ). Thus, it would be in effect a good approximation to calculate
only the 1s , 1D2 and 3s - 3n1 states as done by Brueckner and Gammel. 0 1 Finally we note the most interesting feature, the decisive
saturation effect of the tensor force in the coupled 3s1 - 3n1 states. It is the only contribution to lL which shows definite saturation in
1this density range. The s state, where the force is purely central,0
-
51
will saturate at a higher density comparable to the close packing
of the hard cores.
A central force can act in first order, so that all states
in the Fermi sea can contribute, and saturation occurs only after
the average matrix element becomes small. A tensor force acts only
in second order. Empirically it is a long range force, so can only
scatter to intermediate states through a momentum transfer of order
one inverse fermi. Thus the states which can take advantage of
the tensor force are those in a shell at the top of the Fermi sea.
Since the surface area of the Fermi sea increases less fast than the
volume, the net contribution of the tensor force to the binding will
increase less fast than that of the central force.
If only the 1s and 3s states were important, the mean 0 1
potential energy ~U would lie midway between the S-state curves in
1Figures ? to 9 • However, as we noted above, the D state can be2 regarded as the net result of all the higher states, and its con
tribution is seen to be anti-saturating. Its effect is thus to
-1 (move the saturation point from about ~ =1.3 F where it would 1be with only S-state forces) to the desired region near 1.4 F- •
We can now consider the difference in binding between the
Reid and Bressel potentials. Comparing Figures 7 and 9 we see
that the major difference in the average potential energy curves is
in the 3s - 3n state, where Bressel shows less tendency to saturate1 1 until a higher density. It appears in fact that the Bressel potential
-
52
has a stronger ratio of central to tensor force in this state, in
agreement with the above discussion. This was checked in a rough
manner by computing the phase shifts, for both potentials, using the
central forces alone. Hence, the main difference is that Bressel
-1 3fails to saturate until 1.5 F , due to a weaker s1 tensor force.
But by postponing saturation until the higher density, it gains a
small amount of binding from the other states (which one can say are
1in sum, the s and 1D2). In principle, the gain in binding this 0 way could be greater, but it does not happen to be.
At this point one small defect in the work must be pointed
out. In applying Equation (II-52) to the hard core forces, we ma.de
the error of using '#L{r) f (r) in the final term, rather than f(c). This makes the energies of excited states less attractive than they
would otherwise be, and costs nearly 1 MeV of binding. Thus our
hard core calculatioL is not strictly comparable to the soft core
one where Equation (II-51) could be used directly. On the other hand,
using Day's solution of the Faddeev equations will give compensating t
effects, slig1*ly reducing the binding. These two corrections might
well cancel, and neither is includea as we hesitate to repeat all
the work now.
Another point that should be pointed out here is that f~r
the soft core potential (Bressel-Kerman) while calculating Bethe's
suppression factor f(r) we assumed the particle state momentum to be
-1about 4F • To test if this was reasonable we also calculated f(r)
-
53
at 3F-l and 5F-l and found that what was to be gained in binding
-1 -1 energy at 3F was lost at 5F • Thus the two effects were com
-lpensating and that 4F was a good value to use.
In Figure 10 we show the 3s - 3n reference wave function distortion for the dominant S-channel solution. 'X 20 vanishes
inside the core, since a plane wave has no "small component". This
is calculated for the average relative momentum k = [:3 kF. 0
The Fourier transform of X is small for k' ~ kF because of 00
cancellation between the long range attractive and shi:,:c·t range repul
sive parts of X • This circumstance makes the second order 00
correction from F small. However, it is clear that for X2000 there is no cancellation, and its 92 Hankel transform is negative and quite substantial for k' !::!- kF (Figure 1). X 20 is made
large by the strong tensor force. Thus, it is the tensor force
which makes the second order correction terms so large in the coupled
3s - 3D states. Only in this state is the third order correction
appreciable. The relative smallness of the third order correction
is due to geometrical effects~ only at a few places in the region
of integration do all factors in the integrand take on large values.
In Table X we tabulate GN matrix elements for the Reid
potential, state by state, with statistical w~ights included, at
several sub-nuclear densities. Each line refers to a relative
These results
are of value for further work.
-
In Figure IV one can see the details of the potential
energy and the reference spectrum. This is for the Reid
potential, ~ =1.36 and ~ =o.6·, m* = .949. The curve U is the
potential energy of occupied states using GR. Uc includes the
second and third order corrections evaluated at several k • The0
agreement seems quite good. UM i·s a li"near i t 1 t• b t n erpo a ion e ween
We believe it is wrong to accept the computed
Nvalues of U for ~ ~ kF for the following reason. The very great
Rattraction just above the Fermi surface comes about because (i) G
is modified by the suppression factor f, removing most of the hard
core repulsion and (ii) we take only even waves into account, by
Rajararnan's prescription. At least the second of these effects is
suspect, since Rajarman's argument (16 ) relied heavily on neglecting
all hole-state momenta compared to the momentum of the state con
sidered. This is clearly wrong until kb is at least 2 or 3 ~·
One can also say that the spectrum just above kF will be sensitive to
the long range forces, such as the tensor force. However, for these
an improved treatment of the three body problem is needed, and has
been developed by Dahlblom ( 29). One needs now two suppression
factors f c and fT' and the calculation becomes more involved. Here,
we have simply used Bethe's solution and applied a single f(r) to all
parts of G. We believe that the actual spectrum will be closer to
ifl than to tfi in this region. The spectral correction can be changed one or two MeV by moving uM about, but no attempt was made to do this
-
55
to make the result look artificially better. When a better treat
ment of the three body problem and the energy spectrum is available
it will be a simple matter to correct our result.
In conclusion, we believe the remaining uncertainty in 9ur
results lies in the physical approximations used and not in the
numerical ones. We have just mentioned the necessary improvements
in the treatment of three body effects and in the evaluation of the
energy spectrum near the Fermi surface. The question of the energy
spectrum for particle states is bound up with the convergency of the
theory, and has recently been discussed very nicely by Brandow (15 ).
These effects could certainly be a few MeV in magnitude.
Our result for the compressibility in Table V is of the
same order of magnitude as in Brueckner's calculations. Remarkably,
the soft core potential gives a greater value than the hard core.
This is probably connected with the fact that saturation is not
exclusively a property of the repulsive core. As discussed above,
the Bressel potential saturates at a higher density, where the core
can be playing a more important role in saturation, and this would
account for the higher value of the compressibility coefficient.
In any case, there is no experimental evidence against the present
values which are of the magnitude required recently by Sorensen in
his consideration of the isotope shift.
In Table XI we present the results for the re-arrangement
energy calculated according to the relation
(IV-1)
-
CHAPTER V
SPIN ORBIT FORCE
We have seen that for an infinite system like nuclear
matter the 'G' matrix formalism gives excellent results both for
the Binding Energy and saturation density. An extension of this
procedure to finite systems was initially suggested by Brueckner,
Gammel and Weitzner( 5) (hereafter referred to as B.G.w.). Here
the main term in the effective interaction in a finite nucleus is the
'G' matrix, taken from infinite nuclear matter at that density.
This procedure is commonly known as the "local density approximation".
It has been followed more recently by Kuo and Brown ( ":fJ ) and to some
extent justified by Brandow ( 31 ) •
In nuclear matter, only the central force part of G is
effective for binding and saturation. In a finite nucleus the spin-
orbit, tensor and quadratic spin-orbit forces will also come into
play. Obviously the balance of these various forces will be different
in 'G' than in the original two body force. It is clearly of interest
to know whether, indeed, G is made up of a "strong central-weak tensor"
force, or just what the balance is.
56
-
57
In this section the spin-orbit force is considered
in some detail, and the separation of the effective tensor inter
action is only considered in a general way. Using the effective
spin-orbit force, the one body spin orbit force of the shell model
is evaluated in an approximation similar to that of Blin-Stoyle
( 32 ) • Shell model level splittings are considered, reasonable
values are found and certain problems of current interest are dis
cussed (constancy of splitting among the isotopes of Ca, questions
of the 'shape' of the one-body spin-orbit force of the shell and
optical models).
In view of the satisfactory agreement with experiment
and with other calculations, these techniques should be applied to
the separation of the tensor and other parts of the two body force.
Further, the question of the applicability of the Moszkowski-Scott
(9) method to the spin-orbit force is considered.
Consider the G in its most general form:
(1'.JG//l.):: a+.i.-c(q;+-crz)·!j+-m(
where IT and g:- are axial vectors, ~ and P are polar vectors:
( ~ x ~/)
/-/:: x~'J p " I = * + -k- (V-2)
;: ~· 15 ~ ;
-
58
where ~' ~· are unit vectors in the incident and outgoing directions.
For the spin-orbit force we are interested only in the coefficient c
which is given, by geometrical arguments, to be
I ( i.tf,J -i.
-
59
Then using (V-4a, 5)
Like B.B.P. Equations 6.8 and 6.10 we define
(V-7)
and
;JIM' ut.':r' [l L' l"
Then
I /YJ'0 (V-8)('•'H~ _p_ M u;r lrf-t) I YL I :r >d ~ ?
-
60
where (V-10)
(V-11)
Using
(V-12)
and Equation (V-7) we finally obtain, using the reference G matrix
GR for G
· C (t'' Io fr// ;r "1 1
} C( LI fYl'--n M / .J /Y1 1
)
-
61
As a first use of this equation we apply it to the case
M=M', and sum over M. Then putting k = k ' and using the Clebscho 0 Gordan identity
L c ( L s o M J 7 fYl) c ( L" s o m / .:r /YJ) -= bu·· (Y1 (V-14)
we obtain
(V-15)
which agrees with the corresponding quantity in Equation (6.ll+a) of
B.B.P.
The G-matrix elements given by Equation (V-13) are in the
k-space. It is sometimes useful and necessary to go from the k-space
to the r-space or vice-versa. It is convenient to know how this
affects a given partial wave, where the Fourier Transform induces a radial
Hankel transform. The rules are summarised below:
(1) To go from k space to r space: (4Tt )2 kk' 1 kk'Multiply the matrix elements by •=~(2it3)2 rr' rr'
Then take a double Hankel Fourier transform with respect to 3L (kr) and 8L(k• r') integrating over kk'. The result is the (rr') matrix element of G.
(2) To go from r space to k space. 2(4:t) rr'
Multiply the coefficient by
kk'
-
62
Then take a double Hankel Fourier transform with respect to 3L(kr) and jL(k'r') integrating over rr'. The result is the (kk') matrix
element of G.
While doing these transformations the following identities are
useful:
(a)
(V-17)
(b)
Thus Equation (V-13) and the procedure for going from k space to
r space and vice-versa we know all the matrix elements of G in either space.
Coming back now to the evaluation of 'c' given by Equation(V-3}
we obtain by using Equation (V-13) in it
(V-18)
where
1
l ~ l #/ 'l:i.. { ::
(.iJ (e1l-fl) C (L11-1j:ro)
ff (V-19)
+ c ( l 11 0 I ;r I ) c ( L ,, I 0 I I JI)J
Thus the second term in Equation (V-1) becomes
-
. 'C
=-~ (:!i+y;) ( ~x-f!) 1 L' - -,,. -f< ~ ( 6J)
(V-20)
Using the relation
(V-21)
.L I Bs (JM j 9, 0.,-t.J 11-L~· u,~- u,;..! dLrL Y,_'Coooo; .(?w,,,LL
1lu ;T d(C
-
64
where L = r x k and r = i 'J k• A -
The identity can easily be seen if we take the z-axis along the k'
direction. Putting this result in Equation (V-21) we get B(k ,k ')0 0
as the coefficient of the L.S in the G-matrix as
-k:~ J;;- ~ CSUM J gt (-fo.-t) V-l:' t., t.~ L''
(V-25)
where CSUM = 2·BSIB1 • [ 2L + l
d. ((J.L"+1}(;
-
--
65
C (Llt-1/:ro)::: ::T=L+I
.J = L 0 .::J= L"
-~ J =L-1 -..j.;t-l"-1-11 :r =- l"-1
L(t...+:J.) l ,, +~c (L110/::r1) -- :f;-l+I C ( l 11 / 0 I/JI) = .J: L''+ I
L .:r.:r = -(F· = l " / l { l-tl)
l "- I ,,(L - l}(L+I) -:T=l-1J = l-1
J. (:Jl"r I)L {C).l+1)
(V-27)
It is now a very simple matter to deduce that in Equation (V-26)
(i) if L fi L" CSUM = 0
~L+3 L :;; ::::r- I (V-28)L +-I
. (ii) if L = L" CSUM = JL-1-1- -- L =:TL (Ut)
J L-1 L =::T+I L
-
66
Thus, knowing the L.S coefficient for the T = 1, 0
cases the total coefficient is given by
(V-29)
which for a N = Z nucleus reduces to the usual ~ (T = 1 contribution)
+ 41 (T =0 contribution).
The above expression for the coefficient of the spin-orbit
operator has been obtained in the k space. For a practical application
we must know its form in the r-space. The procedure for such a
transformation has been given on page 61. Before we do this we must
note that for physically interesting cases the coefficient of the
operator is diagonal in the k-space, i.e., k =k ', or else energy0 0
will not be conserved. Let B (r,r') be the r space representation
of the L.S coefficient.
There are two ways in which we can proceed to obtain
B ( r, r 1 ) from B ( k , k ) • Both procedures should at least in 0 0
principle lead to the same result, though in actual practice it is
more convenient and accurate to use one rather than the other. For
physical insights the opposite is the case. We, therefore, mention
-
67
We know from our reference spectrum calculations
Using this relation in (V-25)
C,; Q f>1 J~L (-/,,.t) x:l" (f,,k) J4 f; ( '""') (V-31)
with a similar expression for odd L. Now (see Equation (I~l))
R are simply the diagonal matrix elements in k space of G and can
simply be found by the Bessel Fourier Transform of the wave defect
obtained in the Nuclear Matter calculations. Thus we see that a
suitable combination of the G-matrix elements, the combination being
provided by CSUM (Equation V-26), we can very easily obtain the·
coefficient of the spin-orbit operator in k- 8 pace. To obtain its
r-space representation B (r, r') it is then just a matter of two more Bessel Fourier Transforms according to the prescription on page 61.
-
68
When this procedure is adopted it is a trivial matter to see that
(V-32)
i.e., the coefficient of the spin-orbit operator is symmetrical in
r-space (see Equation V-35).
Explicitly we can write for an N = Z nucleus after taking
{ ~ ("T; 1) + ~ ( /::o) J OF 8 ( "*o,~) 'r~o e-tc r:t
8 (-ko, -Ao) :: ft?\ ( y~.,. /1..2.)[ ( o GI ~ G1 )~ 3 r> m* -R: - fl-~ II+ ""J_ II I, (CO~B)
(V-33)
where (V-34)
Looking at Equation (V-33) it is immediately seen that there is no hard
core conbribution to £3 (k ,k ) since 0 0 )( ~s \ r(c
which is independent of J.
The r space representation for t 1GJ 11 is given byLS
-
Another point that can be easily seen in this procedure
is the absence of the second order correction to the G's. In fact
all G matrix elements which are off diagonal in M have no second order
correction. The proof goes as follows: The second order correction
is given by (See Equation (II-27)):
1G~ =
-
70
Considering, as a special case, the.triplet even states
we can write from Equations (II-41) and (II-42)
//'2 ~ rrt''* ['11' 1"1" fr1 * L- ~" { ·~ ~L'" -r .J ' I ti /II l j .. .., "' tL" J'
~v-en L /.. l
1.:: o.tt J_;J'
Now (YI,, , ,,Jd2.-i
-
71
abandon it in favour of the procedure given below, for reasons mentioned
and also because the k ) kF contribution was masking the k ( ~ 0 0
contribution which, of course, should be the more important one.
The following procedure is less direct and the physical
picture is not clear; but it is more accurate since the total
number of integrations are reduced to a bare minimum.
In Equation (V-25) the only quantity that changes over when
13 (k , k 1 ) ~ B (r,r') is the integral0 0
Consider its Bessel Fourier transform according to the prescription
on page 61. Then
9t. (-kok)13t. (-li.o-'c') vt:· u~,. {J: ~·) dJc'' .~ {(,-'t-') elk() d '/
1
=- 4~4 ff k~' ·: ~ (Jc-k') l{~ l{:.L" (~a'k") 9L(f
-
72
This then is the r space representation of the spin-orbit coefficient
and differs from Equation (32) of B.G.W. by a factor of 2
because of our inclusion of the exchange term. They derived this
equation more directly in the r-space itself.
We thus see that to obtain the r-space 8 (r,r') the number
of integrals have been reduced from three to one. This will certainly
improve accuracy. For a practical application of 8 (r,r') it is
better to delay the integration until the end.
We now wish to consider the spin-orbit splittings of the
shell model potential. As is well known a two body spin orbit force
will lead directly to a one body splitting, while the tensor force
will do so only in second order. According to B.G.w. the second order
effect is negligible; here we consider only L.S because we want to
see if it alone is adequate. We follow Blin-Stoyle (32 ) and others
by considering a "closed shell ±1 nucleon" nucleus, and ask for the one
body force due to the interaction of the odd nucleon with the core.
This procedure is followed because it will lead into the 11usual 11
1 d (>~form ofthe L.S force, which will facilitate comparisons with the r
optical and shell model wells used by other authors. The novel
feature here is that we take into account the nonlocality of the twofoJt.ce
bodyA(G-matrix). Certain approximations have to be made to lead to a
simple result. These are pointed out.
We, therefore, have to consider the expectation of the
spin-orbit term
J3(Jr,..t'> L·S ::: ~ ( '[i-t C!J-) • -'r,z. xp,'2. 6 (-'!:," ,Jr, 1 2 ) (V-45)
-
73
where
= ~, I
- ~... '
( V-1+6)In doing so we closely follow B.G.W.
111 11Suppose that we are considering particle which has a
specified spin state. Then the expectation value of q- summed over2
the spin states of particle 2 will be taken to be zero if the states are
all populated equally with spin up and down. In actual practice, however,
this is not quite correct, since a single particle spin-orbit interaction
splits two states with the same orbital motion, but opposing spins.
This effect is of second order in the spin orbit potential strength and
we shall ignore it since the spin-orbit potential is weak relative to
the central potential. We shall also neglect any possible lack of
spin-pairing in unfilled shells. Consequently our results will hold
at and near closed shells.
To evaluate the expectation value of r 12 x p12, we write ~·~
as
k12 ~ Pn. I (V-47)== r2 ( ~· x t:· + ~2 )( e>- - -'}1 xi:_ .. -~ .. x J?,)
Since P = 0
P:i f3 ( -'t-12 1£,'2) =- f!, 8 (-k,z 1:'1) (V-48)I
(V-49)
-
?4
Consider the expectation value of the first term. This is
11211where C. are statistical weight factors for the states of particleJ
and~. its initial and final wavefunction. The operator p1 of course J operates only on f3 (r r ' ). Because the centre of mass is
12, 12
fixed, i.e., + r 2 = r 1 ' + r I
, we make use of the delta function onr 1 2
it which we have thus far ignored. Then we evaluate the integral over
to obtain
L Ci f ~~ (-:Cz) [ ~1 X J:• 8 ( 11:i , 112 -:J. ( ~. - /c; ll'fi ( -!!2+.ly"';•J (V-51) i ·ol ~·
If 8 were a local function, it would have a delta function on
rl - rl ' • Then r - r 1 ' would not occur in ~j (r2 + r - ' ) and1 1 r 1 we could remove rl x p from the integral. But we know that the non1
locality in 8 is of very short range and that we can imagine that
~. is slowly varying relative to 8 . This will allow us to take J
r 1 x p1 from the integral.
Now consider the expectation value of the second term.
This is
r ci 1~-~ c~z.) r -':.i. )( 1:1 8 c~,2-. 1:1~;J l/J· (-'c:) d~. "1: (V-52) a
-
- -
?5
Since we are integrating over ; and : ' the result of this integration2 2 I
must be proportional to a vector constructed from : and !l • If the1
nonlocality in 8 is of very short range, it is accurate enough to
take this vector to be !i• This approximation is similar to that
a1ready made in removing p1 from the integral of Equation (V-51).
We, therefore, retain only the component of : 2 along : 1 , i.e., we make
the replacement
(V-53)
Then in the expectation value of the spin orbit term we can make the
replacement
k, .Jr.. / z. >~ ~1 X )?,
-
76
Recalling Equation (V-46) this can also be written as
(V-57)
where we have introduced the density correlation function
(V-58)
We now assume that the nonlocality is of short range so that we can
make a Taylor expansion of ~ (r2
, r ' ) about r and retain only the2 2 first two terms of the expansion. Thus using
(V-59)
we can write
(V-6o)
The extra factor of '~' that we have put in compensates for allowing
the operator \J to act on both the coordinates ; 2
• We now make
-
77
a second Taylor expansion between r 2 and r 1 :
(V-61)
and
(V-62)
Substituting these expansions in (V-57)
1 + .!.. ( (.lz-k),,: Jol1 d-'! /3(k,~1 ) ( ~'~?) { (J(/c,) -~ \1-'c,E'(-'l-,) Ol - - ( Q(-\ 1 ) }~I -1:)) 'Qk
1
-
78
Let us now consider a multipole expansion of 8 (r, r I
).
This is given in Equation (V-44). Let us write this equation in the
form
f3 ( .Jc /c'J ::: L 131- (.Jc> -t:') (V-64) l
Within the limits of our nuclear matter calculations 8 1 Cr, r')
is known only for L =1, T = l and L = 2, T =0 states; we took all
other states to be OPEP. Thus there is no experimental information
which denies the assumption that all the multipoles 8 1 are equal;
this assumption leads to a function 8 (r,r') which is a local operator
in the angle variable. An alternative but no more reasonable assumption
would make all the other multipoles zero; this would give maximum
non-lo?ality and have angular dependence P1(Jl..) in place of o(Jl.-Jl.').
(Note: The L =0 multipole is irrelevant since it comes with the
operator L.S). Thus with the above assumptions we can write:
(V-65)
(V-66)
http:131-(.Jc
-
79
where SL , ...S2. 1 are the angles which r, r' make with a given
arbitrary direction.
Let us first consider the T = 1 state. By Equation
(V-63) and (V-66) this becomes
(-".Jv'1-s>lk;JJ,-:,:::: JJI,)
~, . -'E ~ r (-'r,) - .!.. '!- . v... (> ( Jlc,) k.,1 { a ' (V-67)
I I II - -;;-~·'Y,, {JC-'c,)-- (.-\::,-k 1 ).\7.1c ec-'
-
80
1 4 2- - f. Jd h d k. d JL o/. .n..' -k: .lc' f{,r -+,,1c'J ~' · 1 .1c. V (>f.+,) °'1 I< 0 ...t 2. - k,
I
. ( & ( SZ- -.n.') b ( J2 -f- &.')).f.
OY ( Jr., I I I ~ fd k d.t' h" ...-t' 1. 13 (~,.1c'J _!.. d p {V-70)V.
-
81
If we now put these values of B's in Equations (V-68)
and (V-70) we obtain the coefficient of the single-particle spin
oribt operator in r-space. In order to avoid the difficulty experienced
earlier because of lack of knowledge of u for large k it is better to
interchange the order of integration leaving the k-integration to the
last. This will reduce considerably the apparent importance of large k
and thus the contribution from inside the Fermi-sea will be dominant
together with that of immediately outside. Further the number of
integrations will reduce from 3 to 2. When this is done we require
integrals of the type
These can be easily evaluated by introducing exponentially decaying
- f?C.functions of the type e , integrating and then letting l: ~ o.
The general case is a little difficult to work out, but in some particular
Cos2s they are quite simple. This procedure then gives
8 -;.f ~~ 9z (J.x.J d.x
-fl
Jx,J ~~ (f
-
82
Thus combining (V-68) and (V-71) and (V-70) and (V-72), ma.king
the change in integration order as suggested above and then using
(V-73) we shall finally obtain
,( ~ I I v' LS'! k1, ) /I=/ -::;. - qe J d1::2.-k [ f d k , -k, 3 { - V; 0 (-i:) LJ,"( -k,-t_')
(V-74).3 I , I L ' S( 2. , .2 1 2 1'j V, rhJ U, (?
-
These equations give the correct result (namely that of Elin-Stoyle)
in the case of a local two body force. To see this we consider
the Moszkowski Scott separation method. According to this the effective (J.5J
spin orbit part of G is just U- (r) for r ) d and the wave function
U. ~L' -7 ~ L' .t-'lane wav• •
The work follows:
Consider first Equation (V-74). In the Born approximation limit using
vl this becomes
(/,S) , I ( ..Pz:, I V / Jc,) T::-o
(V-77) 00 00
1
16 Jd 3 1I I= 3 ~ ~ ...k I{ /,f ) 0
The factor 6 with v1 is really 2 (2£ +l) and for T = 1 we have onlyong ~ L =1. We now make use of the integral ( 33 )
00
{7( jµ.ff},J.; i)j xA ~ (X-!f} (xv/hdx::: (V-78)0
(7( i»-j-,,.u.,. iJ
- Re ll - ff. < f
-
84
(V-80)
Similarly
after using (V-78) to obtain
(V-82)
Thus
Equations (V-80) and (V-81) correspond to the form of the force given
by Blin-Stoyle.
-
85
Using Equations (V-80, 8.1 and 83) it is now possible
to evaluate the one body spin-orbit force directly using only the
outer part of the two-body spin orbit force. Taking this force
from the Hamada Johnston Potential (Appendix B) we can then show the
one-body force to be
{lS) I 2.. (, • I
( .J.c, Iv /-'c.,);::: - 7f d ·:3. 2..[()•JI0'-13 (I+ µd (V-84)
where d is the distance beyond which the two body force is considered.
We show below that it is equal to or near to the core radius.
Once the effective single particle spin orbit force is
calculated the level splitting is given by b., E = EJ=L~~-EJ=L+Y.!
for an odd nucleon in state (n,,e) by
(v-85)
where -f ne is the radial wave function of the particle and depends
upon the type of well chosen for the nuclear potential.
-
86
Calculations, Results and Discussion
Before we go into the procedure of the calculations and give
any results it should be pointed out that the effective single
particle spin-orbit force consists of the product of two terms:
l d~r dr and its coefficient. Let us call the coefficient by the name VLSCOEF. Thus the spin orbit force is given by
(V-86)
In general VLSCOEF is a function of r, i.e. it is density dependent.
The aim of the present calculations is to find the spin-
orbit splittings in nuclei near to the closed shell ones. The
calculations are in two parts: first is the calculation of VLSCOEF
and the second is using it to find the splittings. Both these are
independent of each other and require quite different assumptions.
The calculation of the actual force is done by using
Equations (V-74, 75, and 76). Equations (V-80, 81 and 83) are also
used if the potential VLS were used directly as the two body force,
rathe:r tnan the G-matrix. The u's are the two body correlated wave-
functions in nuclear matter. In Chapters II and III we have
described how the wave function distortion X. can be calculated in
reference spectrum approximation. The u are then simply given by
(9- X) • Thus knowing the two body poten~ial in the 3p and 3n states it is very easy to calculate the spin-orbit force.
-
One of the main assumptions made so far is that the non
locality of /3 (r,r') is of much shorter range than that of '2 (r,r').
This was found to be true by B.G.W. Preliminary investigations
showed this to be true, but no definite numerical values were obtained
owing to the difficulties experienced in such a calculation as
explained above. The main aim of our calculation, viz. evaluation
of VLSCOEF was reached without explicitly calculating B(r,r'). If the spin-orbit part beyond some separation distance d of
the Hamada-Johnston potential is used, VLSCOEF can be calculated
directly from Equation (V-84). It is found that almost the complete
external potential must be used to get the correct splittings. B.G.W.
made this assumption in their calculations for binding energy effects,
Our justification is provided by Figure 11. Here we have plotted the
two body Hamada-Johnston spin-orbit force as the effective two-body
force calculated in the 'G' matrix approximation. This was obtained
in the following manner after Brandow's suggestion (35) for defining
effective interactions:
From Equations (V-25 and V-28) we see that in momentum space
the effective two body spin-orbit interaction is given by (for T =1 1 L =1 state only and k =k ')
0 0
13 (/ it,u ~.)- ~ u:o-; -U;'u:} ,u,J (V-87)
In Born approximation this reduces to
(V-88)
-
88
Thus the effective spin-orbit interaction is
S( 1. 2. 2. ,_) 3 I I 0 0r v;, u,, + lJ;3 u3, - au, v,- -LJ; u, (V-89)
From Figure 11 it is clear that the effective interaction and the
two body Hamada-Johnston spin-orbit force are very much the same
except at very small distances. It can be said that the "healing"
of the spin-orbit amplitude is remarkably fast. The Hamada-Johnston
spin-orbit force will, therefore, give the same splitting as the
effective one only if almost all of it outside the hard core is used.
From Equation (V-89) we also calculated the effective two body spin-
orbit force using only the spin-orbit part of the Hamada-Johnston
p9tential. This we found to be very close to the one calculated by
using all of the potential. Use of the central part of the
potential gave practically zero value throughout. Thus the tensor
part contribution will be very small and we could conclude that the
effective spin orbit interaction comes mainly from the spin orbit part
of the potential. In Figure 11 the top most curve represents the
difference of the contributions to the two body spin-orbit effective
forces obtained by using the total potential and only the spin-
orbit part of the potential in Equation V-89. This would be the
tensor force contribution. It is seen to be small and vanishes very
fast. These conclusions, however, are at best correct only in first
approximation, since the u's involve the total potential. The
-
above conclusions are true only for the T = 1 case. We have not
investigated the T =0 case; the spin-orbit force here, however,
is small.
One of the interesting points to be studied about VLSCOEF is its
variation with density and the contributions to it from different
regions of the momentum space. These are shown in Figures 12 and
13. Figure 12 gives the variations of the contributions to VLSCOEF
from different regions of the momentum space for T =O,lcomponents, and the total force for N = Z closed shell nucleus. It is seen that the
T =1 part is virtually constant over the whole range of the density, though the contributions from above and below kF change. It is
interesting to note that the variation with ~ is linear and that as
~ increases thereby increasing the size of the occupied region,
the contribution increases by the same amount as the outside sea
contribution decreases. For T = 0 this is not the case. The
outside sea contribution is constant and negligibly small. The
contribution from inside the sea decreases in magnitude as the density
iacreases. The reason for such a variation can be seen from the T =O,
k ( kF contributions given at ~ =l.36 and o.8. This variation is similar to that in the central force where it occurs mainly due to
the 3r>2 states, the 3n1 and 3r>3 are very small and do not change
total 3n much. This variation occurs in 3p states also, but is mutually cancelled by the three states.
-
90
In Table XII we give VLSCOEF at different kF for T =1 and 0 states. The fourth column gives this coefficient for
N = Z closed shell nuclei, while (N - Z)/4A times the number in
fifth column must be added to the corresponding number in the
fourth column to obtain VLSCOEF for N # Z closed shell nuclei.
The numbers are in F3 and must be multiplied by 41.469 to get
them in MeV r. To obtain the spin-orbit force in the form 'constant •
d \' 11 dr we have made several assumptions. This form is useful r since it facilitates comparison with the usual shell model and
optical model potentials. B.a.w. and Brueckner, Lockettand
Rotenberg (7) did not do this. B.G.W. directly used the non
local potential B (r,r') and very ingenj:>usly reduced the integro
differential equation to a pair of differential equations of the
form 'l'" + Ff'' + G'l' = o. Both F and G involve potential functions.
They observed that the spin-orbit part of the G potential was not
localized in the nuclear surface, but due to the presence of the
add5+~_.;;.a~l potential F it is not clear that their work disagrees with
the usual type of analysis.
In Figure 14 we give the VLSCOEF · ! r -dr d (' as well as
d (' 100. -l - • We have seen that the coefficient of ! d ~ is r dr r dr not a constant, but an increasing function of the local density. As
discussed above, this density dependence ~omes from the T =0 part
-
91
of the two body potential. The force is seen to be very similar in
1 d (>shape to - ~ , but peaking about 0.15 F inside the half density
r dr
radius. From optical model analysis of proton scattering data several
authors (36) have found that the radius parameter for the spin-orbit
interaction was approximately .lF less than for the real central
interaction.
Greenless, Pyle and Tang (6) proposed an explanation, based
on the following assumptions, which we shall see are not entirely
justified: (1) the potential well radius squared is estimated to be
the mean square density radius plus the mean square two body force
radius. (2) The two body central mean square radius was taken to be
3i'2 , and the spin-orbit radius zero. However, it is well known that
the long range OPEP does not contribute to the extension of the
nuclear force, because of its exchange character. According to Reid,
the intermediate range two body force is primarily of range of about
2three pion masses, so that its mean square radius is below 1F • On
the other hand, the spin-orbit force has a range of order three to
five pion masses (three in Hamada-Johnston potential) which is not
dramatically shorter than that of the central force. Thus the crucial
steps in the Greenless-Pyle and Tang's argument are not supported.
Our nuclear matter calculations lead easily to a one body
spin-orbit force with a shift of the right magnitude inside the nuclear
density. This is brought about by the density dependence of the
spin-orbit coefficient VLSCOEF (see Figure 12). An argument similar
-
92
to that of the spin-orbit force if made for the central force
would be rather rough, bu